1 INTRODUCTION

Diffractive dissociation in future electron-ion colliders

Anh Dung Le?,

CPHT, CNRS, École polytechnique, IP Paris, F-91128 Palaiseau, France* dung.le@polytechnique.edu

August 2, 2021

Proceedings for the XXVIII International Workshopon Deep-Inelastic Scattering and Related Subjects,

Stony Brook University, New York, USA, 12-16 April 2021doi:10.21468/SciPostPhysProc.?

Abstract

We study diffractive scattering cross sections, focusing on the rapidity gap distribution inrealistic kinematics at future electron-ion colliders. Our study consists in numerical solu-tions of the QCD evolution equations in both fixed and running coupling frameworks. Thefixed and the running coupling equations are shown to lead to different shapes for the rapid-ity gap distribution. The obtained distribution when the coupling is fixed exhibits a shapecharacteristic of a recently developed model for diffractive dissociation, which indicates therelevance of the study of that diffractive observable for the partonic-level understanding ofdiffraction.

1 Introduction

Single diffractive dissociation in electron-hadron collisions is defined to be the scattering processin which the virtual photon mediating the interaction fluctuates into a set of partons part of whichgoes in the final state while the hadron remains intact, leaving a large rapidity gap between thelatter and the slowest produced particle. The rapidity gap characterization of diffractive eventsis manifested in collision experiments by the existence of a large angular sector in the detectorwith no measured particle. Such events were first observed in electron-proton collisions at DESYHERA [1, 2], and their detailed analysis is one of goals of the construction of future electron-ioncolliders [3–5].

Diffractive dissociation in deep-inelastic scattering (DIS) is a process of particular interest.At high energy, the dissociation of the virtual photon can be interpreted as the diffractive frag-mentation of a quark-antiquark dipole state, in the so-called dipole picture of DIS (see [6] for areview). The latter can be described using an elegant formulation [7,8] in the form of QCD non-linear evolution equations of cross sections. In this work, we employ the aforementioned dipolepicture of DIS and numerical solutions to nonlinear evolution equations in both fixed and runningcoupling cases to study the diffractive dissociation of the virtual photon in the scattering off alarge nucleus, with the main attention being on the rapidity gap distribution. Our motivation istwofold. First, the existence of the rapidity gap is the distinguishing feature of diffractive events,and it is used to mark such events in practice. Second, the size of the rapidity gap is, as suggested

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1

2 DIPOLE FORMULATION FOR DIFFRACTIVE DISSOCIATION

by Refs. [9–11], the imprint of the partonic structure of the virtual photon subject to high-energyevolution. Therefore, the rapidity gap distribution is an important observable to measure in futureelectron-ion machines, as it is relevant to the microscopic mechanism of diffraction.

The paper is outlined as follows. In the next section, we will briefly introduce the dipole formu-lation for diffractive dissociation in DIS. In Section 3, the numerical evaluation of the diffractivecross section and of the rapidity gap distribution will be presented. Finally, we will draw someconclusions in Section 4.

2 Dipole formulation for diffractive dissociation

In an appropriate frame, the virtual photon (γ∗) in high-energy DIS converts into a quark-antiquarkpair, which is called onium hereafter, long before the scattering with the nucleus (A). Consequently,the photon cross sections can be expressed as a factorization of the photon wave function and thedipole cross sections in the scattering off a nucleus. Assuming impact parameter independence,the ratio of the diffractive cross section to the total cross section for diffractive γ∗A scatteringevents with a minimal rapidity gap Y0 out of the total rapidity Y reads

σγ∗Adiff(Q

2, Y, Y0)

σγ∗Atot (Q2, Y )

=

∫

d2r∫ 1

0 dz∑

k=L,T ; f |ψfk (r, z,Q2)|2 [2N(r, Y )− Nin(r, Y, Y0)]

∫

d2r∫ 1

0 dz∑

k=L,T ; f |ψfk (r, z,Q2)|2 2N(r, Y )

, (1)

where Q2 is the virtuality of the photon, r is the size of the onium in the two-dimensional trans-verse plane, and z is the longitudinal momentum fraction of the photon taken by the quark. Theprobability densities |ψ f

L,T (r, z,Q2)|2 of the photon-to-onium splitting in the longitudinal (L) andtransverse (T) polarizations are well-known and can be found, for e.g., in Ref. [6].

In Eq. (1), the function N(r, Y ) is the forward elastic scattering amplitude, and Nin(r, Y, Y0) isthe cross section encoding all elastic contributions of the nuclear scattering of a dipole of transversesize r. They both obey the following leading-order (LO) Balitsky-Kovchegov (BK) equation [7,12,13]:

∂YNr =

∫

dpLO(r, r ′)�

Nr ′ +N|r−r ′| −Nr −Nr ′N|r−r ′|�

, (2)

with Nr representing N(r, Y ) or Nin(r, Y, Y0). The LO integral kernel dpLO(r, r ′) is given by

dpLO(r, r ′) =α

2πr2

r ′2|r − r ′|2d2r ′, (3)

where the “reduced" strong coupling α≡ αsNcπ is kept fixed (hereafter denoted by fc). The running-

coupling extension of the BK equation (2) is also available in the literature [8,14–16], with the LOkernel (3) being replaced by an appropriate kernel, depending on the prescription. In the currentstudy, we employ the two following kernels:

(i) the “parent dipole" kernel (denoted by rc:pd) [14]:

dprc:pd(r, r ′) =α(r2)

2πr2

r ′2|r − r ′|2d2r ′, (4)

2

3 NUMERICAL ANALYSIS OF DIFFRACTION

(ii) the Balitsky kernel (denoted by rc:bal) [15]:

dprc:Bal(r, r ′) =α(r2)

2π

�

r2

r ′2|r − r ′|2+

1

r ′2

�

α(r ′2)α(|r − r ′|2)

− 1

�

+1

|r − r ′|2

�

α(|r − r ′|2)α(r ′2)

− 1

��

d2r ′. (5)

In those prescriptions, the strong coupling α(r2) is set to run with size r as in Ref. [14].The BK equation can be solved numerically once the initial condition is set. The amplitude N

is usually initialized at Y = 0 as the McLerran-Venugopalan (MV) amplitude [17,18],

NMV (r, Y = 0) = 1− exp

�

−r2Q2

A

4ln

�

e+1

r2Λ2QC D

��

, (6)

with the nuclear saturation scale at zero rapidity QA assumed to scale as Q2A = 0.26A1/3Λ2

QC D.Meanwhile, the initial condition for Nin is defined at Y = Y0 as

Nin(r, Y = Y0, Y0) = 2N(r, Y0)− N2(r, Y0). (7)

The detailed numerical recipe can be found in Ref. [19].We are interested in the rapidity gap distribution, which is defined as

Πγ∗A(Q2, Y ; Ygap)≡

�

�

�

�

�

1

σγ∗Atot

∂ σγ∗Adiff

∂ Y0

�

�

�

Y0=Ygap

�

�

�

�

�

(8)

in the case of virtual photon-nucleus scattering, and

Πonium(r, Y ; Ygap)≡�

�

�

�

12N∂ Nin

∂ Y0

�

�

�

Y0=Ygap

�

�

�

�

(9)

for the nuclear scattering of an onium.An approach for the latter has been proposed recently [9–11], in the case in which the onium

is much smaller than the inverse nuclear saturation scale at the total rapidity 1/Qs(Y ). The mainconjecture is that diffraction is due to a rare fluctuation in the Fock state of the onium whichcreates a parton with an unusual small transverse momentum. Within this picture, the asymptoticgap distribution reads [9–11]

Πonium∞ (r, Y ; Ygap) = cD

�

YYgap(Y − Ygap)

�3/2

, (10)

valid in the so-called “scaling region” 1� ln 1r2Q2

s (Y )�p

Y and with the constant cD completely

determined [11].

3 Numerical analysis of diffraction

The ratio of the diffractive cross section to the total cross section, or the diffractive fraction forshort, is plotted in Fig. 1 (see Ref. [19] for a complete definition of the setting and parameters).

3

4 CONCLUSION

0 2 4 6 8Q2

0.20

0.25

0.30

0.35

0.40

0.45

0.50(σ

diff

/σ t

ot)γ∗ A

Minimal gap: Y0 = Y/2

fc, Y = 6

fc, Y = 12

rc:pd, Y = 6

rc:pd, Y = 12

rc:bal, Y = 6

rc:bal, Y = 12

Minimal gap: Y0 = Y/2

fc, Y = 6

fc, Y = 12

rc:pd, Y = 6

rc:pd, Y = 12

rc:bal, Y = 6

rc:bal, Y = 12

0 2 4 6 8Q2

Minimal gap: Y0 = Y/4

fc, Y = 6

fc, Y = 12

rc:pd, Y = 6

rc:pd, Y = 12

rc:bal, Y = 6

rc:bal, Y = 12

Minimal gap: Y0 = Y/4

fc, Y = 6

fc, Y = 12

rc:pd, Y = 6

rc:pd, Y = 12

rc:bal, Y = 6

rc:bal, Y = 12

Figure 1: The diffractive fraction for two different values of the minimal rapidity gap Y0and two values of the total rapidity Y as a function of the virtuality Q2. A= 208 is setfor the nuclear mass number.

While the fraction of diffractive events, about 20% − 25% in the fixed coupling scheme and fora low rapidity (Y = 6), is rather close to the values predicted by other studies (for example, seeRef. [20]), the results for other cases seem to overshoot the latter.

Different behaviors of the diffractive fraction can be drawn from Fig. 1. First, the ratio de-creases with Q2, while it grows when boosting to a higher total rapidity Y . These two featuresreflect the dependence of the diffractive fraction on the ratio between the virtuality Q and thesaturation scale Qs(Y ). In particular, the smaller the scale ratio Q/Qs(Y ) is, the more deeply theprocess probes in the saturation region. Consequently, the diffractive fraction gets closer to itsblack-disk limit value (0.5). Furthermore, the running coupling correction produces larger valuesfor the fraction. This can be attributed to the effect of the running coupling to suppress the emis-sion of gluons with large transverse momentum (equivalently, the branching to color dipoles ofsmall transverse sizes in the limit of large Nc) in the evolution of the onium state of the photon.The process is therefore more elastic, and closer to the black-disk limit.

Figure 2 shows the rapidity gap distributions for different scenarios. There is a difference inthe general shape of the distribution between the fixed coupling and running coupling cases. Inparticular, there is a local minium located at a gap value 0 < Ygap < Y , while the distributionis enhanced for the gap values close to 0 or Y . That is not the case when the running couplingeffect is taken into account. For the latter, the distributions appear to grow monotonically whenthe size of the rapidity gap increases. Furthermore, one should notice a notable feature that theshape of the distribution when the strong coupling is fixed is quite similar to the shape predictedby Refs. [9–11] (see Eq. (10)) in the case of onium-nucleus scattering.

4 Conclusion

We have presented numerical predictions for the diffractive fraction and the rapidity gap distribu-tion for kinematics accessible at future elecron-ion colliders, based on the solutions to QCD small-xevolution equations in both fixed coupling and running coupling scenarios. The diffractive events

4

4 CONCLUSION

0 2 4 6Ygap

1.5

2.0

2.5

3.0

Πγ∗ A

×10−2

fc, Y = 6

Q2 = 2 GeV2

Q2 = 5 GeV2

fc, Y = 6

Q2 = 2 GeV2

Q2 = 5 GeV2

0 2 4 6Ygap

rc:pd, Y = 6

Q2 = 2 GeV2

Q2 = 5 GeV2

rc:pd, Y = 6

Q2 = 2 GeV2

Q2 = 5 GeV2

0 2 4 6Ygap

rc:bal, Y = 6

Q2 = 2 GeV2

Q2 = 5 GeV2

rc:bal, Y = 6

Q2 = 2 GeV2

Q2 = 5 GeV2

0 2 4 6 8 10 12Ygap

0.50

0.75

1.00

1.25

1.50

Πγ∗ A

×10−2

fc, Y = 12

Q2 = 2 GeV2

Q2 = 5 GeV2

fc, Y = 12

Q2 = 2 GeV2

Q2 = 5 GeV2

0 2 4 6 8 10 12Ygap

rc:pd, Y = 12

Q2 = 2 GeV2

Q2 = 5 GeV2

rc:pd, Y = 12

Q2 = 2 GeV2

Q2 = 5 GeV2

0 2 4 6 8 10 12Ygap

rc:bal, Y = 12

Q2 = 2 GeV2

Q2 = 5 GeV2

rc:bal, Y = 12

Q2 = 2 GeV2

Q2 = 5 GeV2

Figure 2: Rapidity gap distribution for different values of the total rapidity Y and of thevirtuality Q2. A= 208 is set for the nuclear mass number.

are shown to take a significant fraction in the scattering of a virtual photon off a large nucleus.The shape of the rapidity gap distribution is significantly modified by the inclusion of runningcoupling corrections. Interestingly, the distribution in the fixed coupling case has a shape which isqualitatively similar to the shape deduced from a recently proposed partonic model. This suggeststhat the distribution of the gap size could be a relevant observable for unveiling the microscopicmechanism of diffractive dissociation.

The current work takes into account the running coupling effect, which is the only-knownnext-to-leading correction thus far for the diffractive dissociation. In addition, the knowledge onthe rapidity gap distribution in the onium-nucleus scattering is currently limited to the regime ofasymptotic large rapidity and small onium size in the scaling region. Therefore, to produce betterphenomenological predictions for electron-nucleus collision, further theoretical developments areessential.

Acknowledgements

We would like to thank Stéphane Munier for his valuable comments and for reading the manuscript.

Funding information This work is supported in part by the Agence Nationale de la Rechercheunder the project ANR-16-CE31-0019.

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